I don't have a lot of experience but here are my thoughts: If you're bringing the puppy in the cabin, which is what you want, then the airport terminal setup may be more important than the airline. Some terminals make having a dog easier than others with special areas where you can let them out and so forth. (Think I'd rather be going Sacramento to Boson rather than vice versa).
The other thing is you have to take the pup out of the carrier when you go through security. This can be frightening for the pup so the more used to you they are the easier this will be. Obviously you aren't there but if possible I'd schedule a day so you can just hang out with the pup. Even a day will likely make this less frightening.
Yet another thing is that a four month pup might not be able to hold off peeing for this long so you'll want some extra stuff to handle that. Finally, I wouldn't let them out of the carrier during the flight.
Good luck with your new dog. Exciting!